Exponents for 𝐵-stable ideals

Sommers, Eric; Tymoczko, Julianna

doi:10.1090/S0002-9947-06-04080-3

Exponents for $B$-stable ideals
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by Eric Sommers and Julianna Tymoczko PDF

Trans. Amer. Math. Soc. 358 (2006), 3493-3509 Request permission

Abstract:

Let $G$ be a simple algebraic group over the complex numbers containing a Borel subgroup $B$. Given a $B$-stable ideal $I$ in the nilradical of the Lie algebra of $B$, we define natural numbers $m_1, m_2, \dots , m_k$ which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types $A_n, B_n, C_n$ and some other types. When $I = 0$, we recover the usual exponents of $G$ by Kostant (1959), and one of our conjectures reduces to a well-known factorization of the Poincaré polynomial of the Weyl group. The other conjecture reduces to a well-known result of Arnold-Brieskorn on the factorization of the characteristic polynomial of the corresponding Coxeter hyperplane arrangement.

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